Method, system and medium for controlling manufacture process having multivariate input parameters

ABSTRACT

A method, system, and medium of modeling and/or for controlling a manufacturing process is disclosed. In particular, a method according to embodiments of the present invention includes the step of identifying one or more input parameters. Each input parameter causes a change in at least two outputs. The method also includes the step of storing values of the identified inputs and corresponding empirical output values along with predicted output values. The predicted output values are calculated based on, in part, the values of the identified inputs. The method also includes the step of calculating a set of transform coefficients by minimizing a score equation that is a function of differences between one or more of the empirical output values and their corresponding predicted output values. The method further includes the steps of receiving a new set of values for the identified inputs, transforming the new set of values for the identified input using the set of coefficients, and calculating a set of predicted output values using the transformed input values.

RELATED APPLICATION

[0001] This application claims priority from U.S. ProvisionalApplication No. 60/426,393, filed Nov. 15, 2002, which is incorporatedherein by reference.

FIELD OF THE INVENTION

[0002] The present invention relates to a method, system and medium formodeling and controlling processes. More specifically, the presentinvention relates to modeling and controlling semiconductor-processingequipment that has multivariate input parameters.

BACKGROUND OF THE INVENTION

[0003] In manufacturing products that include precision discrete parts(e.g., microelectronic chips on silicon substrates), controllingmanufacturing processes plays a crucial role. Controlling such processesmay require, among other things, monitoring the characteristics ofmanufactured parts (e.g., processed wafers, hereinafter referred to asoutputs) and adjusting input parameters accordingly. By adjusting thevalues of the input parameters, different types of outputs can beproduced and the characteristics of the outputs can also be controlled.

[0004] For automating the control of the manufacturing processes, amathematical model of the processing equipment can be used. One exampleof such a model is called a predictive model. This model is used topredict the future output values (e.g., the characteristics of products)based on historical information (e.g., input parameter values and thecorresponding output qualities).

[0005] One such predictive model is an offset technique, which isillustrated in FIG. 1. In particular, the values of a number of inputparameters 101 are received by an input/output dependency model 103,which calculates a predicted output value y₁ ^(Pred) 105 based on theinput values. A corrector 109 then compares the predicted value y₁^(Pred) with an actual output value y₁ ^(a) 107 for the given values ofthe input parameters. If the predicted and actual output values aresimilar to each other within a certain range, no change is made to theinput/output dependency model 103. If the predicted and actual outputvalues are different (e.g., outside the range) from each other, thepredictor input/output dependency model 103 is modified by adjusting anoffset value (O₁) 111 based on the magnitude of the difference.

[0006] In equipment that has more than one output, at least some of theoutputs may include mutual (shared) inputs. This means the output valuesof the equipment are not completely independent from each other (e.g.,changing an input to adjust a given output may unintentionally changethe characteristics of other outputs). In a conventional modelingtechnique, each output has its own correction system as if the outputvalues are independent from each other. Because the dependencies betweenthe different outputs are not accounted for by the conventionaltechnique, it does not always lead to accurate predictions. In addition,adjusting one offset of one output can affect other outputs.

SUMMARY OF THE INVENTION

[0007] Embodiments of the present invention advantageously overcome theabove-described shortcomings of the aforementioned techniques. Morespecifically, embodiments of the present invention provide a system,method and medium for controlling semiconductor-processing equipmentthat has multivariate input parameters and outputs.

[0008] Embodiments of the present invention minimize the effects ofoutputs being interdependent from each other. This is achieved byproviding input parameter transformations having transformationcoefficients. The coefficients are obtained by minimizing a scorefunction. This, in turn, allows accurate models to be obtained. Usingthe models, highly precise control of manufacturing equipment isaccomplished.

[0009] In particular, an example method according to embodiments of thepresent invention includes the steps of identifying at least one inputthat causes a change in at least two of a plurality of outputs, storingvalues of the identified inputs and corresponding empirical outputvalues, and calculating and storing predicted output values, based on,in part, the values of the identified inputs. The example method mayfurther include the steps of calculating a set of transform coefficientsby minimizing a score equation that is a function of, in part,differences between one or more of the empirical output values and theircorresponding predicted output values, and calculating one or more inputvalues for one or more desired output values based on, in part, thecalculated set of transform coefficients.

BRIEF DESCRIPTION OF THE DRAWINGS

[0010] The detailed description of the present application showingvarious distinctive features may be best understood when the detaileddescription is read in reference to the appended drawings in which:

[0011]FIG. 1 is a diagram showing a conventional offset model;

[0012]FIG. 2 is a diagram illustrating processing equipment;

[0013]FIG. 3 is a diagram illustrating a model of the processingequipment shown in FIG. 2 in accordance with embodiments of the presentinvention;

[0014]FIG. 4 is a block diagram illustrating various components ofembodiments of the present invention

[0015]FIG. 5 is a flow chart illustrating processing steps ofembodiments of the present invention;

[0016]FIG. 6 is a diagram illustrating a CMP process;

[0017]FIG. 7 is a block diagram representation of an example embodimentof a computer configured to perform embodiments of the presentinvention; and

[0018]FIG. 8 is a diagram illustrating an example of a memory mediumthat can be used for storing computer programs of embodiments of thepresent invention.

DETAILED DESCRIPTION

[0019] Embodiments of the present invention generally provide systems,methods and mediums for creating one or more adaptive process models tomathematically represent multivariate input parameter systems. Thepresent invention is particularly applicable in a manufacturing processsuch as manufacturing and/or processing semiconductor wafers. Inparticular, the present invention relates to modeling techniques as usedby equipment involved in the manufacturing of semiconductor wafers. Ageneral overview of embodiments of the present invention is providedbelow. It will be followed by a specific example implementation of thepresent invention.

[0020] Before discussing embodiments of the present invention, FIG. 2shows a simplified graphical representation of processing equipment 205with input parameters 201 and outputs 203. Examples of processingequipment include etcher tools, deposition tools, chemical mechanicalplanarization (CMP) tools, etc. The processing equipment 205 can includeone or more tools. Depending upon the values of the input parameters201, different processes can be achieved. For instance, in a depositiontool, different types of layers can be deposited on a wafer and/or thethickness of the layer can be varied.

[0021] As a general overview of embodiments of the present invention, inFIG. 3, the processing equipment 205 has a set of input parameters 301,a set of predicted outputs 303, and a prediction model 305 therebetween(replacing the processing equipment of FIG. 2). The overall goal of theprediction model is to minimize differences between the predicted outputvalues and empirically collected output values (i.e., the actual outputvalues). Once the prediction model is optimized (e.g., the differencesbetween the predicted and actual output values have been minimized), themodel can then be used in setting input parameters based on desiredoutput values. In other words, for a given set of desired output values,the model can be used in a reverse fashion to calculate the inputparameter values that would cause output values close to the desiredoutput values. The calculated input parameter values are also known asrecipes.

[0022] In embodiments of the present invention, the step of obtainingthe predictive model can be divided into two steps. The first is totransform the values of the input parameters 301 into transformed inputvalues 307. The second is to use the transformed input values 307 incalculating predicted output values 303.

[0023] With respect to the transformation, input parameter values (X₁,X₂, X₃) along with coefficient vector {right arrow over (P)} aretransformed into (X′₁, X′₂, and X′₃) by transform functions ψ₁ψ₂, andψ₃. Examples of transformation functions include:

[0024] 1) X′₁=PX₁; X′₂=PX₂ (In this example, the value of {right arrowover (P)} is identical for both X1 and X2.)

[0025] 2) X′₁=P₁₁X₁+P₁₂X₁ ²; X′₂=P₂₁X₁+P₂₂X₂ ²+P_(cross) X₁X₂ (In thisexample, P₁₁, P₁₂, P₂₁, P₂₂ and P_(cross) can have different values.)

[0026] The coefficient values are calculated by the steps of: a.collecting historical information on input parameter values and actualoutput values; b. creating a score function based on the collectedinformation; and c. finding the coefficient values that minimize thescore function, S_(p).

[0027] The above steps are described by making references tosemiconductor processing tools. As such, the step of collecting thehistorical information entails a set of data points for processing anumber of wafers. In particular, input parameter values and actualoutput values for a number of wafers that have been processed by theprocessing equipment would be collected. This collection would then beused in the next step of minimizing the score function.

[0028] Here, the score function, S_(p), is:$S_{p} = {\sum\limits_{i,k}{W_{i,k}\left( {y_{actual}^{i\quad k} - {y_{predicted}^{i\quad k}\left( {{\overset{\rightarrow}{X}}^{i^{\prime}}\left( {{\overset{\rightarrow}{X}}^{i},\overset{\rightarrow}{P}} \right)} \right)}} \right)}^{2}}$

[0029] where:

[0030] i—number of wafer;

[0031] k—number of output;

[0032] y_(actual)—an actual output value;

[0033] y_(predicted)—a predicted output value, as calculated based ontransformed inputs for a particular wafer i ({right arrow over(X)}^(′i));

[0034] {right arrow over (X)}^(′i)=(X₁ ^(40 i),X₂ ^(40 i),X₃ ^(′i)) isthe transformed input vector, calculated on the base of the actualinput; and {right arrow over (X)}^(i)=(X₁ ^(i), X₂ ^(i),X₃ ^(i)) forwafer i together with the transformation parameters {right arrow over(P)}. This calculation is performed using the following transformationfunctions:

ψ₁(X₁,X₂,X₃,{right arrow over (P)}): ψ₂(X₁,X₂,X₃,{right arrow over(P)}); and ψ₃(X₁,X₂,X₃,{right arrow over (P)}).

[0035] The next step, as noted above, is to minimize the score S_(p),i.e., to find {right arrow over (P)} values that provide the minimum of$S_{p}\left( {S_{p}\underset{p}{}\min} \right)$

[0036] The above-described steps calculate an optimal {right arrow over(P)} (i.e., a vector of coefficients for input transformation functions)such that the prediction model of the present invention provides theclosest possible predicted outputs to the actual outputs. In aprocessing model with multivariate input parameters, when the score isminimized, the negative effect of the interdependencies between outputvalues on the model accuracy would also be minimized.

[0037] Now turning to describe an example implementation of theembodiments described above, as shown in FIG. 4, the exampleimplementation includes a number of components: an input transformer401, an input-output dependency model 403, a corrector 405 and a storagedevice 407. All these components can be implemented in hardware,firmware, software and/or any combination thereof.

[0038] These components are further explained by also referring to FIG.5. In particular, the historical information (i.e., y_(a) ^(ik),{rightarrow over (X)}^(i)) is stored into the storage device 407. Thecorrector 405 then retrieves the historical information (y_(a) ^(ik),{right arrow over (X)}^(i)) from the storage device 407 (step 501).Since the retrieved historical information contains raw input parametervalues, the information is sent to the input transformer 401 along withcoefficients {right arrow over (P)} (step 503). The coefficient {rightarrow over (P)} can be stored in the storage device 407 or in thecorrector 405.

[0039] The input transformer 401, upon receiving the information fromthe corrector 405, calculates transformed input parameter values {rightarrow over (X)}^(′i) (step 505). Once the transformed input parametervalues are calculated, the input transformer 401 sends the transformedinput values to the corrector 405.

[0040] The corrector 405, upon receiving the transformed input parametervalues from the input transformer 401, sends the transformed inputparameter values to the input/output dependence model 403. Theinput/output dependency model 403 then calculates predicted outputparameter values y_(pred) (step 507). The corrector 405 then calculatesthe score S_(p), and sets a new {right arrow over (P)} (a vector ofparameters of input transformation functions) in order to minimize thescore S_(p) (step 509). These steps can be repeated until an optimum{right arrow over (P)} that yields a minimal score S_(p) is obtained,and return the optimum {right arrow over (P)}. Each time new data isobtained, a new score from new data is created and a new optimum {rightarrow over (P)} value is calculated. This newly calculated vector {rightarrow over (P)} could be used for transforming the input values,meaning: {right arrow over (P)}_(new)≡{right arrow over (P)}_(optimum).

[0041] In embodiments of the present invention, the optimum coefficientscan be combined with the most recent vector such that: {right arrow over(P)}_(new)≡{right arrow over (P)}_(previous)+K({right arrow over(P)}_(optimum)—{right arrow over (P)}_(previous)) wherein K<1.

[0042] As a new set of data points arrives, a new optimum {right arrowover (P)} can be recalculated.

[0043] Once a set of coefficients is calculated, a set of input valuescan be obtained (e.g., a recipe) for a desired set of output values.More specifically, from a set of desired values, a set of transformedinput values, {right arrow over (X)}^(′i), can be obtained by reversingthe predictive model (e.g., the input/output dependence model 403). Thetransformed input values can then be reverse transformed using thecoefficients {right arrow over (P)} to obtain the input value to producethe desired output values.

[0044] In the above-described embodiments, the raw input values aretransformed using the calculated coefficients. The transformation isrequired to account for the dependencies among input parameters asgraphically illustrated in FIG. 6. More specifically, a surface of awafer having five regions with varying degrees of roughness is to bepolished by a CMP process. The goal is to achieve a flat surfacedepicted by a dotted line in FIG. 6. In conventional techniques, oneregion would be polished without regard to the other regions. However,polishing one region can affect the polishing of another region (e.g.,when an offset is applied in region 1 in order to bring the height inregion 1 down to the broken line, the height in region 2 is alsoinfluenced by the changes of region 1). Using the embodiments of thepresent invention, these dependencies are accounted for.

[0045] An example embodiment of the computer in which embodiments of thepresent invention operate (e.g., the various components described inFIG. 4) is described below in connection with FIGS. 7-8. FIG. 7illustrates a block diagram of one example of the internal hardware 713of a computer configured to perform embodiments of the presentinvention. A bus 756 serves as the main information highwayinterconnecting various components therein. CPU 758 is the centralprocessing unit of the internal hardware 713, performing calculationsand logic operations required to execute embodiments of the presentinvention as well as other programs. Read only memory (ROM) 760 andrandom access memory (RAM) 762 constitute the main memory. Diskcontroller 764 interfaces one or more disk drives to the system bus 756.These disk drives are, for example, floppy disk drives 770, or CD ROM orDVD (digital video disks) drives 766, or internal or external harddrives 768. These various disk drives and disk controllers are optionaldevices.

[0046] A display interface 772 interfaces display 748 and permitsinformation from the bus 756 to be displayed on display 748.Communications with external devices, such as the other components ofthe system described above, occur utilizing, for example, communicationport 774. Optical fibers and/or electrical cables and/or conductorsand/or optical communication (e.g., infrared, and the like) and/orwireless communication (e.g., radio frequency (RF), and the like) can beused as the transport medium between the external devices andcommunication port 774. Peripheral interface 754 interfaces the keyboard750 and mouse 752, permitting input data to be transmitted to bus 756.In addition to these components, the internal hardware 713 alsooptionally includes an infrared transmitter and/or infrared receiver.Infrared transmitters are optionally utilized when the computer systemis used in conjunction with one or more of the processingcomponents/stations/modules that transmit/receive data via infraredsignal transmission. Instead of utilizing an infrared transmitter orinfrared receiver, the computer system may also optionally use a lowpower radio transmitter 780 and/or a low power radio receiver 782. Thelow power radio transmitter transmits the signal for reception bycomponents of the production process, and receives signals from thecomponents via the low power radio receiver. The low power radiotransmitter and/or receiver are standard devices in industry.

[0047] Although the computer in FIG. 7 is illustrated having a singleprocessor, a single hard disk drive and a single local memory, theanalyzer is optionally suitably equipped with any multitude orcombination of processors or storage devices. For example, the computermay be replaced by, or combined with, any suitable processing systemoperative in accordance with the principles of embodiments of thepresent invention, including sophisticated calculators, and hand-held,laptop/notebook, mini, mainframe and super computers, as well asprocessing system network combinations of the same.

[0048]FIG. 8 is an illustration of an example computer readable memorymedium 884 utilizable for storing computer readable code orinstructions. As one example, medium 884 may be used with disk drivesillustrated in FIG. 7. Typically, memory media such as floppy disks, ora CD ROM, or a digital video disk will contain, for example, amulti-byte locale for a single byte language and the program informationfor controlling the modeler to enable the computer to perform thefunctions described herein. Alternatively, ROM 760 and/or RAM 762illustrated in FIG. 7 can also be used to store the program informationthat is used to instruct the central processing unit 758 to perform theoperations associated with various automated processes of the presentinvention. Other examples of suitable computer readable media forstoring information include magnetic, electronic, or optical (includingholographic) storage, some combination thereof, etc.

[0049] In general, it should be emphasized that the various componentsof embodiments of the present invention can be implemented in hardware,software or a combination thereof. In such embodiments, the variouscomponents and steps would be implemented in hardware and/or software toperform the functions of embodiments of the present invention. Anypresently available or future developed computer software languageand/or hardware components can be employed in such embodiments of thepresent invention. For example, at least some of the functionalitymentioned above could be implemented using Visual Basic, C, C++, or anyassembly language appropriate in view of the processor(s) being used. Itcould also be written in an interpretive environment such as Java andtransported to multiple destinations to various users.

[0050] The many features and advantages of embodiments of the presentinvention are apparent from the detailed specification, and thus, it isintended by the appended claims to cover all such features andadvantages of the invention which fall within the true spirit and scopeof the invention. Further, since numerous modifications and variationswill readily occur to those skilled in the art, it is not desired tolimit the invention to the exact construction and operation illustratedand described, and accordingly, all suitable modifications andequivalents may be resorted to, falling within the scope of theinvention. For instance, output values can be transformed similar to thetransform performed on the input parameters, and operations can beperformed on the transformed output values similar to those performed onthe transformed input parameters.

What is claimed is:
 1. A method for controlling a manufacturingapparatus, the method comprising the steps of: (a) identifying at leastone input, the at least one input causing a change in at least two of aplurality of outputs; (b) storing values of the identified inputs andcorresponding empirical output values along with predicted outputvalues, wherein the predicted output values are calculated based on, inpart, the values of the identified inputs; (c) calculating a set oftransform coefficients by minimizing a score equation that is a functionof, in part, differences between one or more of the empirical outputvalues and their corresponding predicted output values, wherein thescore equation is:$S_{p} = {\sum\limits_{i,k}{W_{i,k}\left( {y_{actual}^{i\quad k} - {y_{predicted}^{i\quad k}\left( {{\overset{\rightarrow}{X}}^{i^{\prime}}\left( {{\overset{\rightarrow}{X}}^{i},\overset{\rightarrow}{P}} \right)} \right)}} \right)}^{2}}$

where: i—number of wafer; k—number of output; y_(actual)—an empiricaloutput value; y_(predicted)—a predicted output value, as calculatedbased on transformed inputs for a particular wafer i ({right arrow over(X)}^(′i)) {right arrow over (X)}^(′i)=({right arrow over (X)}₁ ^(′i),X₂^(′i),X₃ ^(′i)) is transformed input values in a vector format; {rightarrow over (X)}^(i)=(X₁ ^(i),X₂ ^(i),X₃ ^(i)) for wafer i together withthe transformation parameters {right arrow over (P)}, to therebycalculate an optimal value of {right arrow over (P)}; and (d)calculating one or more input values for one or more desired outputvalues based on, in part, the calculated set of transform coefficients.2. The method of claim 1, further comprising the steps of: collectingadditional empirical data and corresponding input values; calculating anew set of coefficients {right arrow over (P)}_(new); and using the newset of coefficients as the optimal value of {right arrow over (P)}. 3.The method of claim 1 further comprising the steps of: collectingadditional empirical data and corresponding input values; calculating anew set of coefficients as {right arrow over (P)}_(new)≡{right arrowover (P)}_(previous)+K({right arrow over (P)}_(optimum)−{right arrowover (P)}_(previous)), wherein K<1 and {right arrow over (P)}_(previous)is a previously calculated optimal value of {right arrow over (P)}; andusing the new set of coefficients as the optimal value of {right arrowover (P)}.
 4. A system for controlling a manufacturing apparatus, thesystem comprising: (a) means for identifying at least one input, the atleast one input causing a change in at least two of a plurality ofoutputs; (b) a memory device configured to store values of theidentified inputs and corresponding empirical output values along withpredicted output values, wherein the predicted output values arecalculated based on, in part, the values of the identified inputs; (c)means for calculating a set of transform coefficients by minimizing ascore equation that is a function of, in part, differences between oneor more of the empirical output values and their corresponding predictedoutput values, wherein the score equation is:$S_{p} = {\sum\limits_{i,k}{W_{i,k}\left( {y_{actual}^{i\quad k} - {y_{predicted}^{i\quad k}\left( {{\overset{\rightarrow}{X}}^{i^{\prime}}\left( {{\overset{\rightarrow}{X}}^{i},\overset{\rightarrow}{P}} \right)} \right)}} \right)}^{2}}$

where: i—number of wafer; k—number of output; y_(actual)—an empiricaloutput value; y_(predicted)—a predicted output value, as calculatedbased on transformed inputs for a particular wafer i ({right arrow over(X)}^(′i)) {right arrow over (X)}^(′i)=(X₁ ^(′i),X₂ ^(′i),X₃ ^(′i)) istransformed input values in a vector format; {right arrow over(X)}^(i)=(X₁ ^(i),X₂ ^(i),X₃ ^(i)) for wafer i together with thetransformation parameters {right arrow over (P)}, to thereby calculatean optimal value of {right arrow over (P)}; and (d) means forcalculating one or more input values for one or more desired outputvalues based on, in part, the calculated set of transform coefficients.5. The system of claim 4, further comprising: means for collectingadditional empirical data and corresponding input values; and means forcalculating a new set of coefficients {right arrow over (P)}_(new),wherein the new set of coefficients is defined as the optimal value of{right arrow over (P)}.
 6. The system of claim 4, further comprising:means for collecting additional empirical data and corresponding inputvalues; and means for calculating a new set of coefficients as {rightarrow over (P)}_(new)≡{right arrow over (P)}_(previous)+K({right arrowover (P)}_(optimum)−{right arrow over (P)}_(previous)), wherein K<1 and{right arrow over (P)}_(previous) is a previously calculated optimalvalue of {right arrow over (P)}, wherein the new set of coefficients isdefined as the optimal value of {right arrow over (P)}.
 7. A computerreadable medium for storing instructions being executed by one or morecomputers, the instructions directing the one or more computers forpredicting output characteristics of a device produced by amanufacturing apparatus, the instructions comprising implementation ofthe steps of: (a) identifying at least one input, the at least one inputcausing a change in at least two of a plurality of outputs; (b) storingvalues of the identified inputs and corresponding empirical outputvalues along with predicted output values, wherein the predicted outputvalues are calculated based on, in part, the values of the identifiedinputs; (c) calculating a set of transform coefficients by minimizing ascore equation that is a function of, in part, differences between oneor more of the empirical output values and their corresponding predictedoutput values, wherein the score equation is:$S_{p} = {\sum\limits_{i,k}{W_{i,k}\left( {y_{actual}^{i\quad k} - {y_{predicted}^{i\quad k}\left( {{\overset{\rightarrow}{X}}^{i^{\prime}}\left( {{\overset{\rightarrow}{X}}^{i},\overset{\rightarrow}{P}} \right)} \right)}} \right)}^{2}}$

where: i—number of wafer; k—number of output; y_(actual)—an empiricaloutput value; y_(predicted)—a predicted output value, as calculatedbased on transformed inputs for a particular wafer i ({right arrow over(X)}^(′i)) {right arrow over (X)}^(′i)=(X₁ ^(′i),X₂ ^(′i),X₃ ^(′i)) istransformed input values in a vector format; {right arrow over(X)}^(i)=(X₁ ^(i),X₂ ^(i),X₃ ^(i)) for wafer i together with thetransformation parameters {right arrow over (P)}, to thereby calculatean optimal value of {right arrow over (P)}; and (d) calculating one ormore input values for one or more desired output values based on, inpart, the calculated set of transform coefficients.
 8. The medium ofclaim 7, further comprising the steps of: collecting additionalempirical data and corresponding input values; calculating a new set ofcoefficients {right arrow over (P)}_(new); and using the new set ofcoefficients as the optimal value of {right arrow over (P)}.
 9. Themedium of claim 7, further comprising the steps of: collectingadditional empirical data and corresponding input values; calculating anew set of coefficients as {right arrow over (P)}_(new)≡{right arrowover (P)}_(previous)+K({right arrow over (P)}_(optimum)−{right arrowover (P)}_(previous)), wherein K<1 and {right arrow over (P)}_(previous)is a previously calculated optimal value of {right arrow over (P)}; andusing the new set of coefficients as the optimal value of {right arrowover (P)}.
 10. A method for controlling a manufacturing apparatus, themethod comprising the steps of: (a) identifying at least one input thatcauses a change in at least two of a plurality of outputs; (b) storingvalues of the identified inputs and corresponding empirical outputvalues; (c) calculating and storing predicted output values, based on,in part, the values of the identified inputs; (d) calculating a set oftransform coefficients by minimizing a score equation that is a functionof, in part, differences between one or more of the empirical outputvalues and their corresponding predicted output values; and (e)calculating one or more input values for one or more desired outputvalues based on, in part, the calculated set of transform coefficients.11. The method of claim 10, wherein the score function is:$S_{p} = {\sum\limits_{i,k}{W_{i,k}\left( {y_{actual}^{i\quad k} - {y_{predicted}^{i\quad k}\left( {{\overset{\rightarrow}{X}}^{i^{\prime}}\left( {{\overset{\rightarrow}{X}}^{i},\overset{\rightarrow}{P}} \right)} \right)}} \right)}^{2}}$

where: i—number of wafer; k—number of output; y_(actual)—an empiricaloutput value; y_(predicted)—a predicted output value, as calculatedbased on transformed inputs for a particular wafer i ({right arrow over(X)}^(′i)) {right arrow over (X)}^(′i)=(X₁ ^(′i),X₂ ^(′i),X₃ ^(′i)) istransformed input values in a vector format; {right arrow over(X)}^(i)=(X₁ ^(i),X₂ ^(i), X₃ ^(i)) for wafer i together with thetransformation parameters {right arrow over (P)}, to thereby calculatean optimal value of {right arrow over (P)}.
 12. The method of claim 10,further comprising the steps of: collecting additional empirical dataand corresponding input values; calculating a new set of coefficients{right arrow over (P)}_(new); and using the new set of coefficients asthe optimal value of {right arrow over (P)}.
 13. The method of claim 10further comprising the steps of: collecting additional empirical dataand corresponding input values; and calculating a new set ofcoefficients based on the additional empirical data.
 14. The method ofclaim 13, further comprising calculating the new set of coefficientsusing: {right arrow over (P)}_(new)≡{right arrow over(P)}_(previous)+K({right arrow over (P)}_(optimum)−{right arrow over(P)}_(previous)), wherein K<1 and {right arrow over (P)}_(previous) is apreviously calculated optimal value of {right arrow over (P)}; and usingthe new set of coefficients as the optimal value of {right arrow over(P)}.
 15. A system for controlling a manufacturing apparatus, the systemcomprising: (a) means for identifying at least one input that causes achange in at least two of a plurality of outputs; (b) a memory deviceconfigured to store values of the identified inputs and correspondingempirical output values along with predicted output values, wherein thepredicted output values are calculated based on, in part, the values ofthe identified inputs; (c) means for calculating a set of transformcoefficients by minimizing a score equation that is a function of, inpart, differences between one or more of the empirical output values andtheir corresponding predicted output values; and (d) means forcalculating one or more input values for one or more desired outputvalues based on, in part, the calculated set of transform coefficients.16. The system of claim 15, wherein the score equation is:$S_{p} = {\sum\limits_{i,k}{W_{i,k}\left( {y_{actual}^{i\quad k} - {y_{predicted}^{i\quad k}\left( {{\overset{\rightarrow}{X}}^{i^{\prime}}\left( {{\overset{\rightarrow}{X}}^{i},\overset{\rightarrow}{P}} \right)} \right)}} \right)}^{2}}$

where: i—number of wafer; k—number of output; y_(actual)—an empiricaloutput value; y_(predicted)—a predicted output value, as calculatedbased on transformed inputs for a particular wafer i ({right arrow over(X)}^(′i)) {right arrow over (X)}^(′i)=(X₁ ^(′i),X₂ ^(′i),X₃ ^(′i)) istransformed input values in a vector format; {right arrow over(X)}^(i)=(X₁ ^(i), X₂ ^(i), X₃ ^(i)) for wafer i together with thetransformation parameters {right arrow over (P)}, to thereby calculatean optimal value of {right arrow over (P)}.
 17. The system of claim 15,further comprising: means for collecting additional empirical data andcorresponding input values; and means for calculating a new set ofcoefficients {right arrow over (P)}_(new), wherein the new set ofcoefficients is defined as the optimal value of {right arrow over (P)}.18. The system of claim 15, further comprising: means for collectingadditional empirical data and corresponding input values; and means forcalculating a new set of coefficients based on the additional empiricaldata.
 19. The system of claim 18, wherein the means for calculating isfurther configured to use the following equation in calculating the newof coefficients: {right arrow over (P)}_(new)≡{right arrow over(P)}_(previous)+K({right arrow over (P)}_(optimum)−{right arrow over(P)}_(previous)), wherein K<1 and {right arrow over (P)}_(previous) is apreviously calculated optimal value of {right arrow over (P)}, whereinthe new set of coefficients is defined as the optimal value of {rightarrow over (P)}.
 20. A computer readable medium for storing instructionsbeing executed by one or more computers, the instructions directing theone or more computers for predicting output characteristics of a deviceproduced by a manufacturing apparatus, the instructions comprisingimplementation of the steps of: (a) identifying at least one input thatcauses a change in at least two of a plurality of outputs; (b) storingvalues of the identified inputs and corresponding empirical outputvalues; (c) calculating and storing predicted output values, based on,in part, the values of the identified inputs; (d) calculating a set oftransform coefficients by minimizing a score equation that is a functionof, in part, differences between one or more of the empirical outputvalues and their corresponding predicted output values; and (e)calculating one or more input values for one or more desired outputvalues based on, in part, the calculated set of transform coefficients.21. The method of claim 20, wherein the score function is:$S_{p} = {\sum\limits_{i,k}{W_{i,k}\left( {y_{actual}^{i\quad k} - {y_{predicted}^{i\quad k}\left( {{\overset{\rightarrow}{X}}^{i^{\prime}}\left( {{\overset{\rightarrow}{X}}^{i},\overset{\rightarrow}{P}} \right)} \right)}} \right)}^{2}}$

where: i—number of wafer; k—number of output; y_(actual)—an empiricaloutput value; y_(predicted)—a predicted output value, as calculatedbased on transformed inputs for a particular wafer i ({right arrow over(X)}^(′i)) {right arrow over (X)}^(′i)=(X₁ ^(′i),X₂ ^(′i),X₃ ^(′i)) istransformed input values in a vector format; {right arrow over(X)}^(i)=(X₁ ^(i),X₂ ^(i),X₃ ^(i)) for wafer i together with thetransformation parameters {right arrow over (P)}, to thereby calculatean optimal value of {right arrow over (P)}.
 22. The medium of claim 20,further comprising the steps of: collecting additional empirical dataand corresponding input values; calculating a new set of coefficients{right arrow over (P)}_(new); and using the new set of coefficients asthe optimal value of {right arrow over (P)}.
 23. The medium of claim 20,further comprising the steps of: collecting additional empirical dataand corresponding input values; calculating a new set of coefficients as{right arrow over (P)}_(new)≡{right arrow over (P)}_(previous)+K({rightarrow over (P)}_(optimum)−{right arrow over (P)}_(previous)), whereinK<1 and {right arrow over (P)}_(previous) is a previously calculatedoptimal value of {right arrow over (P)}; and using the new set ofcoefficients as the optimal value of {right arrow over (P)}.